Multi Space Reduced Basis Preconditioners for Large-Scale Parametrized PDEs

Abstract

In this work we introduce a new two-level preconditioner for the ecient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" ne grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built upon a new Multi Space Reduced Basis (MSRB) method that we introduce for the rst time in this paper, where a reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the exible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are wellsuited to perform a single iteration, by addressing the error components which have not been treated yet. The Krylov iterations employed to solve the resulting preconditioned system targets small tolerances with a very small iteration count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in dierent large scale computational settings related to parametrized advection diusion equations and compared with the current state of the art algebraic multigrid preconditioners.

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

“…Indeed, the goal is to implement a preconditioner which (i) can easily handle PDE problems within a prescribed class, although showing remarkable differences in the physical regime, and that (ii) can be efficiently built for several instances of the PDE problem, relying on common structures that can be pre-computed and stored. A more in-depth analysis, which goes beyond the goal of this paper, is reported in [1].…”

“…, L − 1, Q MSRB,k is nonsingular and we refer to its inverse as Q −1 MSRB,k (µ) = P MSRB,k (µ). Moreover, the error e (k) (µ) can be bounded as (see [1] for the proofs)…”

“…By relying on high-fidelity solvers such as the finite element (FE) method [2][3][4] and spectral methods [4,5], a discretized problem is obtained through a (Petrov-)Galerkin projection onto a subspace V h of dimension N h , 10 4 ≤ N h ≤ 10 10 , thus yielding a parametrized linear system A h (µ)u h (µ) = f h (µ), (1) to solve for each new parameter instance. Here u h (µ), f h (µ) ∈ R N h and A h (µ) ∈ R N h ×N h is the stiffness matrix; µ ∈ D ⊂ R p , p ≥ 1, is a vector of p parameters.…”

“…The multi space reduced basis (MSRB) preconditioner, recently proposed in [1], takes advantage of the parametric dependence of the PDE to devise a preconditioning technique tailored for the whole class of linear systems obtained when the parameter vector varies in the selected parameter domain. Such a technique relies on a two-level approach which combines multiplicatively a fine, nonsingular preconditioner (e.g.…”

“…

AbstractWe analyze the numerical performance of a preconditioning technique recently proposed in [1] for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner.

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